Let m be the smallest positive integer such that the coefficient of ${x}^{2}$ in the expansion of ${\left(1 + x\right)}^{2} + {\left(1 + x\right)}$ ^{3} + \ldots + {\left(1 + x\right)}^{49} + {\left(1 + mx\right)}^{50} $is \left(3n + 1\right){}^{51}{C}_{3}$ for some positive integer n.
Then the value of n is

Question

Let m be the smallest positive integer such that the coefficient of

{x}^{2}

in the expansion of

{\left(1 + x\right)}^{2} + {\left(1 + x\right)}

^{3} + \ldots + {\left(1 + x\right)}^{49} + {\left(1 + mx\right)}^{50}

is \left(3n + 1\right){}^{51}{C}_{3}

for some positive integer n.
Then the value of n is

Quantrex Academy · Accepted Answer

Correct answer: 0

Coefficient of x ^ 2 in the expansion of (1 + x) ^ 2 + (1 + x) ^ 3 + (1 + x) ^ 49 + (1 + mx) ^ 50 is ^ 2 C _ 2 + ^ 3 C _ 2 + ^ 49 C _ 2 + ^ 50 C _ 2 m ^ 2 = (3n + 1) ^ 51 C _ 3 ⇒ ^ 3 C _ 3 + ^ 3 C _ 2 + ^ 49 C _ 2 + ^ 50 C _ 2 m ^ 2 = (3n + 1) ^ 51 C _ 3 (Use ^ n C _ r + ^ n C _ r + 1 = ^ n + 1 C _ r + 1 ) ⇒ ^ 50 C _ 3 + ^ 50 C _ 2 m ^ 2 = (3n + 1) ^ 51 C _ 3 (50.49.48)/(6) + (50.49)/(2) m ^ 2 = (3n + 1)(51.50.49)/(6) m ^ 2 = 51n + 1 must be a perfect square By trial ⇒ n = 5 and m = 16 ( M, n ∈ N ) ⇒ n = 5

Let m be the smallest positive integer such that the coefficient of...

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